Universal Quantification for Self-Organized Criticality in Atmospheric Flows

A. Mary Selvam

Indian Institute of Tropical Meteorology, Pune - 411 008, India

Proceedings (Poster) Conference on Patterns,Nonlinear Dynamics and Stochastic Behaviour in Spatially Extended Complex Systems, October 24-28,1997, Budapest, Hungary

(Retired) email: mailto:amselvam@gmail.com

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http://amselvam.webs.com/index.html

Atmospheric flows exhibit selfsimilar fluctuations on all scales (space-time) ranging from climate (kilometers/years) to turbulence (millimeters/seconds) manifested as the fractal geometry to the global cloud cover pattern concomitant with inverse power law form for power spectra of temporal fluctuations. Selfsimilar fluctuations implying long-range correlations are ubiquitous to dynamical systems in nature and are identified as signatures of self-organized criticality. Standard models in meteorological theory cannot explain satisfactorily the observed self-organized criticality in atmospheric flows. Also, mathematical models for simulation and prediction of atmospheric flows are nonlinear and computer realizations give unrealistic solutions because of deterministic chaos, a direct consequence of finite precision round-off error doubling for each iteration of iterative computations incorporated in long-term numerical integration schemes used for model solutions (References ). An alternative non-deterministic cell dynamical system model [A. Mary Selvam, Can. J. Phys. 68 (1990) 831] predicts the observed self-organized criticality as a direct consequence of quantumlike mechanics governing flow dynamics.The model concepts show that the centripetal acceleration representing the inertial mass of eddy circulation follows inverse square law form analogous to Newton's third law for planetary motion.

Model Concepts

Atmospheric flows is a representative example of turbulent fluid flows.The model is based on the concept that spatial integration of enclosed small scale (turbulent eddy) fluctuations result in organized large eddy circulations. The eddy energy spectrum therefore follows statistical normal distribution characteristics according to the Central Limit Theorem. Therefore, the square of the eddy amplitude, i.e., variance represents the statistical normal probability density distribution.Such a result that the additive amplitudes of eddies, when squared, represent probability densities is observed in the sub-atomic dynamics of quantum systems such as the electron or photon. Atmospheric flows therefore follow quantumlike mechanical laws. The root mean square (r.m.s) circulation speed W of large eddy of radius R is then given in terms of enclosed small scale eddy circulation speed w_* and radius r as

(1)

The square of the eddy amplitude W²represents the kinetic energy E given as (from Eq.1)

E=Hn_R

(2)

where n_R ( proportional to 1/R ) is the frequency of the large eddy and H is a constant equal to

for growth of large eddies sustained by constant energy input proportional to w_*² from fixed primary small scale eddy fluctuations. Energy content of eddies is therefore similar to quantum systems which can possess only discrete quanta or packets of energy content hn where h is a universal constant of nature ( Planck's constant) and n is the frequency in cycles per second of the electromagnetic radiation.
The relative phase angle between large and turbulent eddies is equal to r/R and is directly proportional to W² (Eq.1). The phase angle therefore represents variance and also there is progressive increase in phase with increase in wavelength. The above relationship between phase angle, variance and frequency has been identified as Berry's Phase in the subatomic dynamics of quantum systems. Writing Eq(1) in terms of the periodicities T and t of large and small eddies respectively we obtain

(3)

Equation(3) is analogous to Kepler's third law of planetary motion, namely, the square of the planet's year (period) to the cube of the mean distance from the Sun is the same for all planets and results in inverse square law analogous to the Newton's inverse square law for gravitation. The model is similar to a superstring model for subatomic dynamics [M. Kaku, New Scientist 18 Jan., 1997, 32] which unifies quantum mechanical and classical concepts and incorporates gravitational forces along with nuclear and electromagnetic forces. The cumulative sum of centripetal forces in a hierarchy of vortex circulations may result in the observed inverse square law form for gravitational attraction between inertial masses (of the eddies).

WAVE-PARTICLE DUALITY IN CLOUD FORMATION

The apparent paradox of wave-particle duality in microscopic scale quantum systems is however physically consistent in the context of macroscale atmospheric flows since the bi-directional energy flow structure of a complete atmospheric eddy results in the formation of clouds in updrafts regions and dissipation of clouds in downdrafts regions.The commonplace occurrence of clouds in a row is a manifestation of wave-particle duality in the macroscale quantum system of atmospheric flows. The above-described analogy of quantum-like mechanics for atmospheric flows is similar to the concept of a subquantum level of fluctuations whose space- time organization gives rise to the observed manifestation of subatomic phenomena, i.e., quantum systems as order out of chaos phenomena [G.Grossing, Il Nuovo Cimento 103B (1989) 497]. H. E. Puthoff [Phys. Rev.A. 39, 2333 (1989)] has also put forth the concept of "Gravity as a zero-point fluctuation force" ( Quantum fluctuations of empty space ).The vacuum zero-point fluctuation (electromagnetic) energy manifested in the Casimir effect is analogous to the turbulent scale fluctuations whose spatial integration results in coherent large eddy structures.